Articles of functional analysis

Sot convergence of a net

The following are exercises of Conway’s operator theory: I proved both exercises, but I confused about this point that in exercise 8, $T_i\to 0$ (sot), so based on exercise 6, $T_i^2 = T_i.T_i\to 0$ (sot) while it’s not true. Is not it a counterexample for exercise 6? Thanks.

Injection from non-separable to separable subspaces

Let $\Gamma$ be an uncountable set (possibly of cardinality $\aleph_1$). Is there an injective bounded linear operator $T\colon c_0(\Gamma)\to X$, where a) $X$ is some separable Banach space b) $X=c_0$? Thank you in advance. EDIT: This might be useful as well: Johnson and Zippin proved that each quotient of $c_0$ is in fact its subspace. […]

About functions of bounded variation

I got the following the following idea in one of the articles that I’m reading. It goes this way. Let $X$ be a Hausdorff topological vector space and let $\mathcal{D}$ be the family of all divisions of the compact interval $[a,b]$. For function $g:[a,b]\to X$ and each $D\in \mathcal{D}$, we write $$V(g,D)=(D)\sum [g(v)-g(u)]$$ where $$D=\{[u,v]\}$$ […]

Is the limit of compact operators again compact?

Let $(T_n)_{n \in \mathbb{N}} \subset \mathcal{L}(\mathcal{X}, \mathcal{Y})$ where $T_n$, $n \in \mathbb{N}$, is compact. Now, assuming that $(T_n)_{n \in \mathbb{N}}$ has a limit $T \in \mathcal{L}(\mathcal{X}, \mathcal{Y})$ with respect to the operator norm, e.g. $\|T_n-T\| \rightarrow 0$. Is $T$ also compact in the case where $\mathcal{X}$ and/or $\mathcal{Y}$ are Hilbert spaces, $\mathcal{X}$ and/or $\mathcal{Y}$ are […]

Norm of linear functional; can we take supremum over dense subset?

If $V \subset H$ are Hilbert spaces and $V$ is dense in $H$, is it true that for $f \in H^*$, $$\lVert f \rVert_{H^*} = \sup_{v \in V} \frac{|f(v)|}{\lVert v \rVert_V}?$$ So I mean can we just take the supremum in the definition of the norm of $f$ over the dense subset?

If $E$ is not complemented in $X$, is $E \oplus \{0\}$ not complemented in $X \oplus Y$?

Question: Let $X$ be a Banach space, and let $E \subset X$ be a closed subspace such that $E$ is not complemented in $X$. Does it follow that $E \oplus \{0\}$ is not complemented in $X \oplus Y$, where $Y$ is some other fixed arbitrary Banach space? I ask this question because I want to […]

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 4: How to show boundedness?

Let $f_1$, $f_2$ be the functionals defined on the normed space $C[a,b]$ of all continuous functions defined on the closed interval $[a,b]$ with the maximum norm be defined as follows: $$f_1(x) \colon= \max_{t\in[a,b]} x(t), \; \; \; f_2(x) \colon= \min_{t\in[a,b]} x(t) \; \; \; \forall x \in C[a,b].$$ Then $f_1$ and $f_2$ are not linear […]

A nonlinear version of the Riesz isomorphism

The present question regards the proof of the following theorem which is found in Adams’ Sobolev spaces, §2.30 – 2.33. Here $(\Omega, \mathcal{M}, \mu)$ denotes some arbitrary measure space and $L^p$ always stands for $L^p(\Omega)$. Riesz Representation Theorem Let $1<p<\infty$ and let $p’=p/(p-1)$. For any $v \in L^{p’}$ denote $L_v$ to be the linear functional […]

characterisation of compactness in the space of all convergent sequences

I go through a proof of the following. Let $(\ell_1,d)$ be the metric space of all sequences $x = (\xi_i)_{i \in \mathbb{N}}$ with $\sum_{i=1}^{\infty} |\xi_i| < \infty$ and the metric $$ d(x,y) = \sum_{i=1}^{\infty} |\xi_i – \eta_i|, \qquad x = (\xi_i), y = (\eta_i). $$ Theorem: A subset $M$ of $l_1$ is totally bounded (pre-compact) […]

Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Fix some positive integers $L$ and $k \leq L$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= (r_i)_{i=1}^{N}$ such that $L – \sum\limits^N_{i=1} r_i < k$ and all the $r_i \geq k$. Prove (or disprove) that the minimum of $$ \mathcal{H}(r) […]